(Sections)
Determining What Constitutes Existence and Entities in Physics
One difficult question in our discussion is considering what we mean by the range of "our world," specifically the limitations or range of existence. Intuitively, "our world" refers to the physical world– one that is objective and independent of us. On one hand, logical positivism suggests a distinction between empirical and non-empirical, emphasizing the limitations of the scientific method that the meaning in propositions is only its empirical consequences. Proposing such a distinction between abstract and concrete entities, as proposed in logical positivism, seems arbitrary and unacceptable. The physical existence is objective and not dependent on us, making it arbitrary to propose criteria for ontological commitment based solely on human observation abilities. On the other hand, the terms "objective" and "independent" themselves are ambiguous, and there are theoretical limitations on where, what, and how any observer can ever observe in principle. These limitations are objective, yet they also imply the necessity of accounting for our role as observers within the system. I will explore the different types of theoretical limitations of observation and their potential influence on our consideration of criteria for ontological commitments.
The first theoretical limitation is the region of causal horizon, a clear distinction between what is in principle empirical and what is non-empirical, but it does not imply a physical boundary. This kind of theoretical limitation appears to be the constraint on how far information can travel to a determined frame of reference, due to the constant speed of light. According to Einstein's general relativity, there exists a light cone that represents the maximum distance light can travel from each frame of reference. This implies the maximum range of information that can be received in principle, but does not imply that the choice of frame of reference has consequences on the system itself. In other words, the limitations of light cones only implies a fact about location, instead of suggesting any heterogeneity between the region inside and the light cone from outside. Furthermore, not only is there no clear boundary separating empirical from non-empirical consider General Relativity, but our universe seems to suggest a structure without boundaries. This speculation is supported by our measurement of the quantum fluctuations of empty space, which have a positive vacuum energy. According to the law of gravity and our observation of the universe's expansion, it is not likely if our universe had a boundary. This example does not provide a reason to separate what is empirical and non-empirical in our criteria for ontological commitment.
There is another type of theoretical limitation of observation, related closely to Heisenber uncertainty principle, which remains more sophisticated invoicing the role of observer. In particle physics, to measure is to use extremely high energy to smash it. Thus, the act of measuring or observing at such a small scale appears to be a factor intrinsic to the system being described. Certainly there are ways in which we don’t have to always use high energy to smash the particles to measure in a direct way. Experiments such as those involving the Electric Dipole Moment (EDM) yield high precision for indirect measurements, which has resulted in a precise alignment with our understanding of the existence of quarks and neutrinos. However, there seems to be a theoretical limitation on how small we can observe, which is the Planck length. As far as we know, attempting to observe with greater precision beyond Planck length would require an amount of energy that would collapse into a black hole, with further observations (energy) resulting in progressively larger black holes. Consequently, whether there is a form of existence smaller than the Planck length remains unanswered. The Planck length limitations are closely related to the Heisenberg uncertainty principle, which suggests that the more precisely one property is measured, would result in the less precise the measurement of other properties. Such features of the limitation of our observation seem to be an objective theoretical limitation of the world. At the same time it indicates the action of observation is a factor within the systems that needs to be described to yield a correct narration– observation is part of the “what is.” In light of these considerations, the concept of a world independent of us becomes ambiguous. This kind of theoretical limitation of observation is conceptually different from the first one, which solely pertains to the observer's frame of reference.
The third example involves both the physical limitations of observations as well as demanding a need for description within the causally disconnected frames of reference. In detail, the ER=EPR conjecture is often time considered resolved by distinguishing between information and correlations– the information from an entangled pair must follow a classical path to be received– the speed of how fast information can travel is still limited by the speed of light. However, considering ER=EPR as two entangled independent systems introducing a mechanism like a traversable wormhole, there remains a problem since it is theoretically possible when an observer jump into one side of the wormhole and gain information about the otherside, even though they can never exit again. This scenario introduces a mystery where classical quantum mechanics currently lacks a description to explain what would happen from the perspective inside the Einstein–Rosen bridge, which is, in principle, non-empirical to us as external observers. Experimentally verifying or understanding what occurs from the inside would be impossible unless someone were to enter the wormhole and never return. Such modes of description could potentially play a fundamental role in resolving the challenges posed by the incongruity between our theories of quantum mechanics and general relativity. This configuration suggests a need to describe regions that are causally disconnected from us (and therefore non-empirical) in order to account for the theoretical unity and explain the empirical predictions.
These examples demonstrate on the one hand that there should not be a definitive divide between the empirical and non-empirical in our consideration of criteria for ontological commitment to existence. Such a division implies a privileged way of thinking that designates only the information following the classical path or causally connected to us as the "correct" version of the physical world. On the other hand, the consideration of our observations and measurements remains indispensable to understanding the instance of events. In particular, when facing the second theoretical limitations, it seems to be a decision to make of whether limiting the range of existence solely to what we have observed yields a clearer ontology, resolving the question of whether one should commit to the existence of non-possible possibilia.
That is to say, the question of whether there is a fact of the matter regarding the existence smaller than the Planck length, is based on our best scientific interpretation that yields the most explanatory power. Therefore, I suggest that there could be three levels of ontological commitment to the existence of an entity or a kind (starting from the strongest): 1. Whether it could be observed in any frame of reference, including frames of reference that are causally disconnected from us. 2. Whether it could exert influences on our other observations, that is, whether it can be indirectly observed from any frames of reference. 3. Whether it is necessarily non-reducible to exert explanatory power over empirical observations.
So far, my investigation has focused on determining what it is reasonable for us to commit to in terms of existence with respect to science. This commitment does not depend on the language or logical systems we use, nor does it rely on our observational abilities. Instead, it depends on the best scientific interpretations of the empirical world, aiming to establish the most effective model that possesses explanatory power. Such reliance on the scientific framework is distinctively different from what Quine suggests, which is to take all linguistic objects that are indispensable for the total account of true propositions of IBE and quantifying over terms in first-order logic, granting them as “existing”. It is also different from Carnap's logical positivism, which distinguishes between our acceptance of a system of entities (which is not reliant on science) and the postulates of actual entities that ought to be supported by empirical evidence.
The second distinct question is about objects or entities. The question of entities pertaining to science is a distinct question from whether there is a fundamental level of entities in the physical world. Our challenge is only concerned with whether there is a reasonable standard to distinguish entities from the rest within the range of existence.
One reasonable standard to determine entities, or individuality, in light of the consideration of what counts as particles, is to think about whether it can be separated from its environment arbitrarily far away and remain stable even though it does not entirely capture our intuition of entities. A counter example would be, for instance, quarks cannot exist in isolation and separate themselves arbitrarily far away from their environment. They are bound by strong nuclear forces, and the amount of energy required to separate them would have to melt them. But we tend to consider quarks and gluons as entities, since they are distinct enough to separate from its environment conceptually, which is countable and carries distinct properties. But then there is a secondary question is to which extent these entities are stable, in terms of how long they exist. Intuitively, if an entity exists for an arbitrarily short amount of time, it would be unreasonable to accept its existence. A reasonable standard could be that it exists for a duration longer than the time it takes for light to pass through its diameter. Otherwise, it would not be too reasonable to accept its existence as an entity. In this sense, according to the actual quality of the world, the standard of individuality is not a question that has strictly binary answers. Instead, it falls along a spectrum of degrees. In other words, the condition of individuality includes whether an entity can be removed from its context and background arbitrarily far away and remain intact, or whether it carries a unique property that determines its identity and allows for its conceptual separation from its background, that is, whether it is countable. Another factor is the reasonable duration of time an entity lasts before decaying, yielding a meaningful notion of individuality.
Determining the Existential status and Entities in Mathematics
Mathematics, as we know, encompasses a vast realm that extends far beyond what is necessary for physics. This suggests that we cannot determine the ontological status of mathematics solely based on our best scientific interpretations. I will then discuss the problem of mathematics in relation to the debate of Platonism and Nominalism.
In the debate of Platonism vs Nominalism, a core question is to understand what the disagreements are about, specific on and around the meaning of “existence” and “object”. The Platonism intuition stems from the fact that we often conceive of numbers and sets in ways that mirror our linguistic expressions involving material objects– for instance, the addition of five apples to five apples resulting in ten apples. The distinction from such intuition lies in the fact that apples are concrete, while numbers are abstract. The nominalism intuition stems from the entire field of mathematics demonstrates pronounced contingency based on the different choice of axioms. For instance, in the case of Axiom of Choice, if we start with a model of the Axiom of Choice (AC) and progressing to an expanded model through forcing, the latter model equally satisfies AC. However, the conceivable scenario arises wherein initiation with a model V with AC, transitioning to a model V [G] through forcing, yields a model V subset W subset V [G] where AC fails within W. In such instances, the determination of whether AC is true is contingent upon the specific axiomatic system employed rather than reflecting an inherent aspect of the external world. The core of nominalism lies in the absence of determinate facts adjudicating the trueness of axioms– different axioms give rise to distinct mathematical frameworks. Thus, the mathematical enterprise is viewed as a game with initial rules, and various mathematical objects are considered mere terms rather than actual entities with intrinsic substance. Nominalism posits the perspective that mathematical objects (objects, relations, and structures) either do not exist, or their existence is not necessary for making sense of mathematics.
Platonism puts forth the view that mathematical objects, such as those in arithmetic, exist in the sense that they are abstract– universal and necessary. On the other hand, Nominalism holds the intuition that mathematical objects are merely constructions. If it is the case that mathematical objects are indeed necessary and universal, it would provide stronger support for Platonism. Conversely, if it is shown that mathematical objects are not necessary and universal, it would lean more towards supporting Nominalism.
It seems that most people agree arithmetic is necessary for mathematics. From my view, the challenge lies in how we account for their seemingly necessity and special status in mathematics. One of the main features entaling its special status is that it has enough structure to allow the encoding of other structures inside of it. Which means, once we have arithmetic, we can use it as a base to compare other theories. Euclidean geometry, in contrast, is too simple to allow arbitrary encoding. This suggests that in some sense, arithmetic is a standard to determine how complex or functional a system can be. Mere addition shares a completely different level of complexity than having both addition and multiplication. Which suggests that when someone says we cannot do math without arithmetic, what they are actually saying is that we need the amount of complexity that can be encoding arithmetic to do interesting math. Meanwhile, there are other systems that can also be used and are equivalent to arithmetics (after specifying the appropriate arithmetics), such as the Zermelo-Fraenkel set theory without the axiom of infinity. And certainly, there are different arithmetic systems other than Peano Arithmetic, such as Tarski Arithmetic, Robinson’s Q, and could follow Harvey Friedman, to try finding “natural axioms” that are independent of Peano Arithmetic. In other words, the definition of arithmetic entirely depends on the axiomatic rules we choose. Therefore, we can attribute the necessity of arithmetics to the fact that it forces a complexity level that encodes structures, instead of the so-called metaphysical necessity.
So the next question is the question of identity– whether every element in a model has a unique definition that pins it down uniquely. Still this question does not have a binary, definitive answer of either "yes" or "no." Instead, the answer can vary depending on the context. The collection of elements does have a unique definition, for instance, the natural numbers with just the successor relation. But it does not have a unique definition if the case involves a structure where we start with natural numbers and replace each element with an infinite set of indistinguishable elements. This means that whether there is a distinct, unique definition of each mathematical element also depends on the choice of axioms.
In our discussion of necessity and identity, the dependence of axioms does not necessarily lean towards either Platonism or Nominalism. Platonism suggests that mathematical objects exist independently of us, are abstract– universal and necessary. Yet it does not fully account for how the quality of universal and necessary in mathematical objects are contingent on the chosen set of axioms. On the other hand, Nominalism posits that mathematics is made up of mere terms with predetermined rules, but it fails to adequately address the objective identities of mathematical objects once we agree on the axioms.
Since the ontological status of mathematics is contingent upon the set of axioms we adopt, the determination and acceptance of the ontological status of mathematical objects can only be discussed and evaluated in relation to those specific axioms. As there is no objective or universally agreed-upon criterion for determining the truthfulness of axioms, the question of whether mathematical objects exist independently of the chosen axioms lacks a singular, definitive answer. A better criterion for ontological commitment to mathematical objects should therefore diverge from that applied to physical objects, as it is not contingent upon the features of our physical world, but rather on the selection of axioms. A parallelism is drawn that the laws of the physical world should determine the ontological status of physical existence, as the choices of axioms do the same for mathematics. That is to say, our commitment to the ontology of mathematics should only be with respect to the axioms (or initial contents, in the case of reverse math), similar to how our commitment to the ontology of physics should only be with respect to the actuality of our world.
I have shown that in physics, the determination of what constitutes existence, be it entities or categories, does not adhere to a clear-cut definition, as the notion of clarity often fails to align with actuality. I also emphasize the significance of specifying the specific range of existence under consideration, which is determined by our best interpretations. Similarly, in the realm of mathematics, the existential status of mathematical entities appears to rely solely on the chosen axioms, encompassing factors such as necessity (complexity level), unique identities, and individuality.
Reference:
Carnap, Rudolf. Empiricism, Semantics, and Ontology. Bobbs-Merrill, 1950.
Clarke-Doane, Justin. Mathematics and Metaphilosophy. Cambridge University Press, 2022.
D. Jafferies, personal communication, Dec 11, 2023
N. Ackerman, personal communication, Dec 10, 2023
Putnam, Hilary. Philosophy of Logic. Routledge, 2011.
Quine, W. V. Word and Object. MIT Press, 2013.
Quine, W. V. From a Logical Point of View 9 Logico-Philosophical Essays. Harper & Row, 1963.
Quine, W. V. On What There Is. Bobbs-Merrill, 1948.
Quine, W. V. Two Dogmas of Empiricism. Longmans, Green & Co, 1951.
Quine, W. V. “Carnap and Logical Truth.” Logic and Language, 1962, pp. 39–63, https://doi.org/10.1007/978-94-017-2111-0_5.
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